If a⃗ = 2î − ĵ + 2k̂ and b⃗ = −î + 2ĵ − k̂, find (i) the unit vector along 2a⃗ + 3b⃗, and (ii) a vector of magnitude 6 opposite in direction to 2a⃗ + 3b⃗.

1. 1 · Write what is given and what is to be found. Given: a⃗ and b⃗. To find: the unit vector along 2a⃗ + 3b⃗ and a vector of magnitude 6 opposite to it.
2. 2 · Write the rules. Unit vector = vector/|vector|. A vector of length k opposite to c⃗ is −kĉ.
3. 3 · Find 2a⃗ + 3b⃗.
   • 2a⃗ = 4î − 2ĵ + 4k̂ and 3b⃗ = −3î + 6ĵ − 3k̂
   • 2a⃗ + 3b⃗ = î + 4ĵ + k̂
4. 4 · Find the unit vector. |î + 4ĵ + k̂| = √(1 + 16 + 1) = √18 = 3√2, so the unit vector is (î + 4ĵ + k̂)/(3√2).
5. 5 · Find the vector of magnitude 6, opposite. −6 × (î + 4ĵ + k̂)/(3√2) = −√2(î + 4ĵ + k̂), since 6/(3√2) = 2/√2 = √2.
6. 6 · Check and box the answer. (i) (î + 4ĵ + k̂)/(3√2); (ii) −√2(î + 4ĵ + k̂). Check: |−√2(î + 4ĵ + k̂)| = √2 × 3√2 = 6.

Scoring tip: Simplify 6/(3√2) to √2. It makes the final check easy and looks neat.